Proposition

Let $V$ be a finite dimensional vector space with basis $B=\{v_1,\cdots,v_n\}$. For a linear map $T\in\mathscr{L}(V)$, the following statements are equivalent: 1. The matrix of $T$ with respect to $B$ is upper-triangular 2. $T(v_{k})\in\mbox{span}(v_1,\cdots,v_k)$ for $1\le k\le n$ 3. $\mbox{span}(v_1,\cdots,v_k)$ is $T$-invariant for $1\le k\le n$